Considering the path \(\mathcal {P}\)  in (REF ),
we propose a reference vector field for the closed-loop
dynamics that achieves the LFC problem objective. This vector
field is inspired from the ILOS path following laws for
underactuated marine craft control
(see, e.g., [1]}). In ILOS-based guidance schemes, the
cross-track error of the moving
object to the desired path is minimized while the controlled
object is pointing at a moving target point on the desired path
(see Figure REF (a)). In addition to the cross track-error to \(\mathcal {P}\) ,
ILOS-based guidance laws incorporate the integral of the cross-track error using a
dynamic variable \(s\) . In the presence of disturbances that drive the
microswimmer away from its desired path, embedding the integral
compensation dynamics via the dynamic variable \(s\)  will build up a corrective action
in the reference vector field.
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